The Thermodynamic Formalism and the Central Limit Theorem for Stochastic Perturbations of Circle Maps with a Break
Received 30 November 2021; accepted 04 May 2022
2022, Vol. 18, no. 2, pp. 253-287
Author(s): Dzhalilov A., Mayer D., Aliyev A.
Let $T\in C^{2+\varepsilon}(S^{1}\setminus\{x_b^{}\})$, $\varepsilon>0$, be an orientation preserving circle homeomorphism with rotation number $\rho_T^{}=[k_1^{},\,k_2^{},\,\ldots,\,k_m^{},\,1,\,1,\,\ldots]$, $m\ge1$, and a single break point $x_b^{}$. Stochastic perturbations $\overline{z}_{n+1}^{} = T(\overline{z}_n^{}) + \sigma \xi_{n+1}^{}$, $\overline{z}_0^{}:=z\in S^1$ of critical circle maps have been studied some time ago by Diaz-Espinoza and de la Llave, who showed for the resulting sum of random variables a central limit theorem and its rate of convergence. Their approach used the renormalization group technique. We will use here Sinai's et al. thermodynamic formalism approach, generalised to circle maps with a break point by Dzhalilov et al., to extend the above results to circle homemorphisms with a break point. This and the sequence of dynamical partitions allows us, following earlier work of Vul at al., to establish a symbolic dynamics for any point~${z\in S^1}$ and to define a~transfer operator whose leading eigenvalue can be used to bound the Lyapunov function. To prove the central limit theorem and its convergence rate we decompose the stochastic sequence via a Taylor expansion in the variables $\xi_i$ into the linear term $L_n^{}(z_0^{})= \xi_n^{}+\sum\limits_{k=1}^{n-1}\xi_k^{}\prod\limits_{j=k}^{n-1} T'(z_j^{})$, ${z_0^{}\in S^1}$ and a higher order term, which is possible in a neighbourhood $A_k^n$ of the points $z_k^{}$, ${k\le n-1}$, not containing the break points of $T^{n}$. For this we construct for a certain sequence~$\{n_m^{}\}$ a series of neighbourhoods $A_k^{n_m^{}}$ of the points $z_k^{}$ which do not contain any break point of the map $T^{q_{n_m^{}}^{}}$, $q_{n_m^{}}^{}$ the first return times of $T$. The proof of our results follows from the proof of the central limit theorem for the linearized process.
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